Hi, I need to find the domain and rnge of the following:
1. X= 1/2 (cos theta + theta sin theta)
Y=1/2 (sin theta - theta cos theta)
total guess here, but here goes
Domain cos theta = +/- 1 and domain sin theta = +/- 1 so domain theta sin theta = 0- +/-infinity
so domain of X= 1/2 (cos theta + theta sin theta) = 1/2 - +/- infinity
Then range with similar logic of Y=1/2 (sin theta - theta cos theta) = 0 - +/- infinity
Last, domain if x = cot theta
Thank yu
Range using similar logic
As usual, the domain is the set of values for x where the function is defined.
x = 1/2 (cosθ + θsinθ)
domain is all real numbers.
range is all real numbers, since θsinθ oscillates between θ and -θ.
y = 1/2 (sinθ - θcosθ)
same domain and range as for x
This graph is a pair of spirals, starting at (1,0) and expanding outwards, clockwise for t<0, counterclockwise for t>0
To find the domain and range of the functions X and Y, we need to analyze the possible values that theta can take.
Let's start by considering the domain of X, which is given by the expression X = 1/2 (cos theta + theta sin theta). Since the cosine function has a domain of all real numbers, we don't need to worry about the cos theta term. However, the theta sin theta term has some restrictions.
To find the domain of theta sin theta, we need to consider the individual domains of the sin theta and theta functions. The sine function has a domain of all real numbers, so again, we don't have any restrictions from the sin theta term. The theta function, on the other hand, has a domain of all real numbers as well.
Therefore, the overall domain of X is the same as the domain of theta, which is all real numbers.
Now, let's move on to the range of Y, which is given by the expression Y = 1/2 (sin theta - theta cos theta). The sin theta term has a range of -1 to 1, so we have the range of sin theta as [-1, 1]. The theta cos theta term doesn't impose any additional restrictions on the range.
Hence, the overall range of Y is [-1/2, 1/2].
Finally, you mentioned finding the domain if X = cot theta. The cotangent function has a domain of all real numbers except when cos theta = 0, because cotangent is undefined for those values. In other words, the domain of X = cot theta is all real numbers except for theta values that make cos theta equal to zero.
To summarize:
Domain of X: All real numbers
Range of Y: [-1/2, 1/2]
Domain of X = cot theta: All real numbers except theta values where cos theta = 0.